RESEARCH PAPERS: Mechanisms Papers

A generalized matrix for the description of differential displacement in three dimension is developed with constraint equations for both synthesis and analysis of mechanisms. Application of the matrix is shown in generating the fixed and moving screw axes surfaces of the motion of various space mechanisms.

A general closed form method of planar kinematic synthesis, using complex numbers to represent link vectors, is applied to the synthesis of a geared five-bar linkage for function generation. Equations are derived and a computer program is developed to yield several solutions. Angular displacements of the input, a cycloidal crank, and the output, a simple follower, are used as linear analogs of the independent and the dependent variables, respectively. A method is demonstrated for six precision conditions (three first, three second-order precision conditions). Numerical examples are included, and the structural error of these geared five-bars are compared to that of optimized four-bar linkages generating the same functions.

The motion of certain spatial mechanisms with up to five links and containing at least one ball-and-socket joint can be visualized and analyzed by considering the intersections of three-dimensional surfaces generated by the mechanism. In this paper these solids and their intersections are represented by electronic analog computer, which can fully simulate in this way the motion capabilities of each link of the mechanism, and plot their relative displacements.

A dual dynamic equation, based on dual vectors and screw calculus, is formulated here to provide a concise analytical tool for the study of the dynamics of rigid members in any complex mechanical system. In this paper the equation is applied to inertia force analysis of an RCCC mechanism of general proportions; bearing reactions and inertia torque are obtained in closed form dual-number expressions. Such analytical expressions are more susceptible for geometric interpretation and well adapted for digital computation.

An analytical method is presented for designing spatial four-link RCCC and RCRC crank-rocker mechanisms with optimum transmission angle. Mathematical relationships are derived for the computation of kinematic parameters of the designed mechanisms. These relationships are utilized to produce design charts.

A technique is developed for applying a previously derived shaking moment optimization theory for force-balanced four-bar linkages. To this end, a standard linkage configuration is introduced. In addition, the RMS dimensionless moment is formulated as a means of comparing the moment characteristics of different mechanism families.

Rolamite geometries having rectangular and diamond cutouts are presented as examples illustrating the use of the geometry analysis computer codes for accurately predicting force traces. The resultant traces are quite different from those predicted by the simple energy analysis, and agree well with experimental results.

A least-square theory for the optimization of the shaking moment of fully force-balanced inline four-bar linkages, running at constant input angular velocity, is presented. It is shown that the optimum conditions of partial moment balance are given by certain link mass distribution ratios. These optimum ratios are found to be functions of link length ratios only.

The concept of screw coordinates is developed in terms of motor algebra, and applied to the kinematics and statics of rigid bodies, in particular to the computer-aided motion analysis of spatial mechanisms. The laws of the composition and transformation of screw coordinates and their application to the kinematics and statics of rigid bodies are developed. These results form the basis for the development of numerical methods for the kinematic analysis of spatial mechanisms.

The basic concepts of screw coordinates described in Part I are applied to the numerical kinematic analysis of spatial mechanisms. The techniques are illustrated with reference to the displacement, velocity, and static-force-and-torque analysis of a general, single-degree-of-freedom spatial mechanism: a seven-link mechanism with screw pairs (H)7 . By specialization the associated computer program is capable of analyzing many other single-loop spatial mechanisms. Numerical examples illustrate the results.

The geared five-bar linkage is the foundation for a function generation problem meeting specifications for 5 multiply separated positions and containing 4 free design parameters. The four-bar linkage is shown to be a member of this class of mechanisms. Design examples of rarely treated functions are given with the quality of the generated approximation. Suggestions are made in terms of the 4 design parameters to assist the designer in obtaining good results.

The principle of point-position-reduction technique, which is used for position-synthesis of planar mechanisms, is extended to synthesize spherical four-link and spatial four-link RCCC mechanisms for four precision positions.

The method of determining the static force and torque distributions in space mechanisms by use of the 3 × 3 screw matrix is presented. Transmissivities in favor of rotation and translation, and angles of transmission in space mechanisms are defined. Transmissivities are used to show why some space mechanisms have no dead center positions. In the process, the dual equilibrium equations, one for each link, are written in dual vector form, then they are solved simultaneously for the dual force components. Explicit expressions for forces and torques in the RCCC and RSSR space mechanisms are obtained. Torque distribution in the spherical four-bar mechanism is reduced from the results for the RCCC space mechanism. The numerical example considers a symmetrical crank-rocker RSSR space mechanism, where the output torque is due to the load inertia. Variations in the input torque, output torque, mechanical advantage, and transmission angle for one cycle are given for both geometric inversions.

Revisions of the Denavit-Hartenberg symbolic notation are proposed which extend its use to the entire domain of rigid link mechanisms. The extended notation provides a clear separation of the pair variables and the invariant parameters of a mechanism and thus provides a framework in which higher pairs can be systematically modeled. The new symbolism can be used directly with the existing matrix methods of kinematic analysis, and a numerical scheme is presented to reduce the task of data collection for these methods.

This optimization process uses only output data received from an analog simulation of the four-bar mechanism. The complexity of the analytical expression for the coupler curve is illustrated and explains the necessity of such a complex method to the curve synthesis. The minimization of the matching error between the desired and the obtained coupler curve is achieved by a combined relaxation and gradient method, and could also be used for other problems which require optimum solutions. An attempt is made to compare the random method with the presented procedure.

The magnitude of the perturbations to the steady-state rotation of most mechanisms is difficult to control with present methods. This paper shows an alternative method of reducing the magnitude of these oscillations. The analysis is carried out for a single slider-crank mechanism found in a tandem compressor. The analysis shows that an order of magnitude reduction in these oscillations is obtainable.

The motion coefficients and design methods for a wheel rolling externally or internally to a fixed wheel are given in Part I for linkage constraint satisfying up to 6 multiply separated positions. Part II provides design equations and graphical procedures in terms of the cubic circle and center point curves based on four symmetrical, multiply separated positions of the moving wheel. The important special case of an angular-circular transformer mechanism with constant mechanical advantage is treated in detail for four positions in Part III.

The general form of the mobility equation is presented, which predicts the number of inputs required to drive a multiloop mechanism having redundant freedoms, passive freedoms, different general constraints in different loops, and groups of loops having over-closing constraints. Redundant freedoms, passive freedoms, general constraints, over-closing constraints are described, and the use of the mobility equation is illustrated through numerous examples. A method of identifying the class of a mechanism is also given.

This paper describes a method of analyzing the kinematic characteristics of cams, with profiles represented by an eight-term Fourier series and with oscillating-type followers. Two kinds of cam follower systems, viz., roller follower and flat-face follower systems, are used. In general the method applies to any cam whose profile can be expressed in terms of its radius being a function of the rotation angle [r = f(θ)].

For certain tasks, four-bar linkages may not provide needed accuracy and/or structural characteristics. To overcome this, one or more bars may be added to the coupler with geared pairs to maintain a “one-degree-of-freedom” system. Utilizing complex numbers and matrix methods, a general geared n-bar function generator is developed in this paper. The computer program devised synthesizes four-bar linkages to approximate the desired function and increases the number of links by one if specifications for accuracy and other requirements are not met. Synthesized linkages are analyzed and then optimized by way of minimizing a multidimensional objective function. As a practical illustration of the n-bar theory, geared five-bar, one-loop function generators are designed to simulate the dynamic response of a two-degree-of-freedom vibrating system.

The RSRC space mechanism is synthesized to generate constrained and unconstrained screws of the form ψ̂(θ) = ψ0 (θ) + εψ1 (θ), where θ is the input parameter. A variational principle is used to determine the dimensions of the optimum mechanism, where the dual error in the generated screw is minimized at a discrete set of design positions whose number vary from three up to 24 for each component of the screw. Parameters of constraints are introduced to eliminate the equations of constraints. These parameters provide that the related constraining conditions are satisfied exactly. They are also used to generate the rotation interval and advance-to-return time ratios in quick-return mechanisms exactly. Seven numerical examples are given, which include the design of 4R plane mechanism to generate screw of zero pitch having odd order instantaneous dwell: RSRC mechanisms to generate screws of infinite pitch, constant pitch, and variable pitch having odd and even order instantaneous dwells; RSRC (or RSCR) doublecrank mechanism having dwell in the output rotation, and quick-return mechanism.

The kinematic structure of epicyclic drives has been investigated with the aid of Boolean algebra. The correspondence between the graph representation of the structure, the mechanism, and the form of the displacement equations has been derived. A canonical graph representation has been given. A method, believed to be novel, is described for the determination of the algebraic displacement equations by inspection, directly from the kinematic structure. The theory can be applied similarly to dynamic analysis and computer-aided sketching and animation.

The paper describes the equivalent of Roberts’ law for Watt-1 mechanisms. It is shown that two floating links of the mechanism may be in turn generated by more than one mechanism of this type. Another investigation reveals that the Watt-1 six-bar (coupler-) curve of degree 14 will be produced by an infinite number of cognates of the same form.

This paper is a further development of the kinematic problem presented in our 1967 paper [1] in which we have obtained the transmission functions for different orders of plane systems with many degrees of freedom. This paper establishes the corresponding system of differential equations of motion beginning with these functions. The purpose of this paper is to facilitate computer programming. Our study is based on the work of R. Beyer [2, 3] and is the first original addition to his papers. A second original contribution to Beyer’s theories is the deductive method of solution, from general to particular, which we have, incorporated in our work. Beyer concluded that the cases having two or three degrees of freedom can be considered as particular solutions to the results obtained.

A novel numerical method to determine the numbers of teeth in the driver and driven gears in a compound gear train is presented as an algorithm for computer-aided design applications. A brief review of available methods for double-pair gear sets is made, and compared to the proposed algorithm from the viewpoints of precision, computational efficiency, and ease of programming. In comparison, we now have for the first time completely eliminated dependency on tables of prime factors and conjugate fractions, and/or the ingenuity of the designer (albeit an interesting intellectual challenge), and reduced the problem to a direct, straightforward calculation as a self-contained algorithm easily programmed by the designer himself. The arbitrary irrational value of 1/π is used in a numerical example; sixteen candidate gear sets are obtained having an overall ratio within ±5 micro-units. The triple-pair gear set design problem is considered, and an extended form of the algorithm derived. This allows, also for the first time, the systematic design of a triple-pair gear set, and obviates heuristic or cut and try methods. The same value of 1/π is used in an illustrative example, and the results presented in tabular form.

Freudenstein’s equation for planar four-bar function generators correlates input and output crank positions implicitly in a scalar expression, with coefficients that are functions of link proportions. Applying this approach to planar geared function generator linkages leads to nonlinear systems of algebraic equations. By the principle of superposition taken from the matrix theory of linear systems and by Sylvester’s dyalitic elimination, closed form solutions are obtained. When the geared linkages are changed into the planar four-bar by setting certain link lengths equal to zero, the generalized equations derived here specialize to Freudenstein’s well-known equation. Results of computer programs for synthesis and analysis based on this theory are tabulated.

A model of a general mechanical system, comprising a pneumatic system coupled to a linkage mechanism, is developed. The dynamic system behavior is studied using the digital computer as a design tool to determine the effect on the performance of changing design parameters. Within practical limitations, models of this kind have been used to achieve optimum design. Novel methods are used to treat the two major components of the system. The pneumatic system is modeled using an equivalent flow area which is a function of the time-dependent pressure ratio. Differential equations used to solve for the transient flow in the reservoir/piping/piston system are derived here. The mechanism driven by the pneumatic system consists of multiple series chains of four-bar linkages. The first- and second-order kinematic ratios required in the dynamics are computed from new explicit expressions derived here. Frictional losses, impact, and flexibility of the mechanism are included in the dynamic model. The nonlinearity of the differential equations arises from: (a) the kinematics, (b) drag forces depending on velocity squared, (c) magnetic forces depending on time squared, and (d) the strong nonlinearity of the time-dependent pneumatic system. The system of equations is solved numerically to obtain the record of pressure, temperature, and leakage in the pneumatic system and the travel of the mechanisms versus time. Good agreement is obtained between the theoretical solution and actual tests.

Using (3 × 3) matrices with dual-number elements, closed form displacement relationships are derived for a spatial five-link R-R-C-C-R mechanism. The input-output closed form displacement relationship is an eighth degree polynomial equation. A numerical example is presented.

Using the principle of similarly varying triangles, the existence of the coupler cognate of space five-link RHHHH and HHRHH mechanisms is established for the general case in which the coupler point is arbitrarily located in the plane of one of their coupler links.

The structural analysis of space linkages with two general constraints has been conducted by using Franke’s condensed notation. Linkages with mobility one and mobility two with up to three loops are considered. Censuses of chains with eight, nine, eleven and twelve links and a case study describing a practical application of these chains are presented.

Spatial mechanisms with up to five links and containing at least one ball joint can be solved by considering the intersection of the three dimensional surfaces which can be generated by portions of the mechanism. This paper presents a method whereby stereoscopic pairs of the surfaces can be drawn using an electronic analog computer and the figures viewed as if they were three dimensional. This results in a far better visualization of the surfaces. In many cases it can be seen what the type of intersection between surfaces is. This determines the gross motion of the mechanism, the limits of its motion, and an estimate of its transmission properties.

An investigation of the vibratory bending response of the elastic connecting rod of a slider-crank mechanism is presented. The response of the system is found to be dependent upon five dimensionless parameters. These are classified as the length, mass, damping, external piston force, and frequency parameters; and the effects of these parameters on the response of the system are investigated. The dynamic behavior is described as graphs of nondimensionalized deflection versus crank angle. Solutions of the linear as well as the nonlinear forms of the equations are included.

The equations of the undulating elastica describing the static behavior of an end-loaded flexible strip include forces, elastic properties, dimensions, and some parameters which are not physically measurable. This paper presents techniques whereby the designer can determine reasonable first approximations to the parameters contained in the equations of the undulating elastica. These first approximations are particularly useful for the iterative solution of the equations for the force or motion analysis of flexible link mechanisms containing one or more members that undergo large elastic deflections. Examples of the application of these techniques are presented.

An improved mathematical model of a mechanism is described whereby the elastic deformations of its links are determined. A method of solving for these deflections is presented along with examples of the elastic behavior of both plane and space mechanisms. Finally, the dynamic forces in a mechanism are discussed and one example given.

A method is described using screw coordinates for the dynamical analysis of systems of pair-connected rigid bodies, such as plane and three-dimensional mechanisms, including gyroscopes. The use of screw coordinates leads to general, linear, and computationally efficient numerical procedures, which are illustrated by a dynamic analysis of a constrained spatial RSKR (or R4 S) linkage.

For the Instrument Engineer involved in the design of mechanisms that transmit power under boundary lubricated conditions, little information is available on which to base fine pitch gear load capacity and life. This paper discusses a gear test program and the development of a rating formula for the surface loading of these gears, and in particular those made from stainless steel and aluminum and stocked by precision gear manufacturers.

The four-link kinematic chain is studied in an effort to establish a bounded region to limit the chain link lengths based upon transmission angle inequality constraints. The resulting constraint limitations are displayed graphically with the chain and their algebraic curve properties are delineated. Simple approximations of these complex loci are developed to facilitate practical application.

Using the principle of stretch rotation and the principle of similarly varying triangle, existence of coupler cognates of eight-link mechanisms with ternary and quaternary links is established. It is shown that there are two additional stationary foci; each of these stationary foci yields an infinite number of cognate mechanisms for a given source eight-link mechanism.

Using the principle of stretch rotation and the principle of similarly varying triangle, the existence of coupler cognates of eight-link mechanisms with ternary links and double joints is established. It is shown that there are two additional stationary pivots; each of these stationary pivots yields an infinite number of cognate mechanisms for a given source eight-link mechanism.

A mathematical model of an elastic mechanical joint with clearances has been formulated and the dynamical equations of motion derived (Part I). The model, which we have called an Impact Pair, is basic to the determination of the dynamical response of mechanical and electromechanical systems with clearances, including determination of dynamic force amplification, frequency response, time-displacement characteristics, and other dynamic characteristics. Whenever possible, the results for the impact pair under various operating conditions are illustrated by graphs, which may also offer some insight into the behavior of clearance-coupled systems.

The theory developed in Part I has been applied to the determination of the dynamic response of the Impact Pair under various operating conditions. Simulation techniques, as well as approximate methods used in control theory, have been used for this purpose.

A set of expressions termed the generalized d’Alembert force is determined for application to two and three-dimensional dynamic analysis of discrete, nonlinear, multifreedom, constrained, mechanical dynamic systems. These expressions greatly simplify the task of developing a correct set of second order differential equations of motion for mechanical systems which are nonlinear because of large deflections or other geometric effects. They apply to both constrained and unconstrained mechanical systems via the method of Lagrange equations with constraint. The two-dimensional version of the expressions has been successfully applied in a type-varient computer program for the dynamic analysis of mechanical networks, and example problems simulated with this program are discussed.

Three finitely separated positions of a rigid body determine circles which pass through the three homologous positions of each point. In this paper we determine the points in the body for which the radii of the associated circles have stationary values. In particular, we determine the global minimum, i.e., the point (or points), for which the associated circle has the smallest radius.

Application of the theory on cognate six-bars of Watt’s type makes it possible to link the well-known inversion mechanisms through mechanical cognation. It is proved that the inversors of Peaucellier, Hart, and Sylvester are cognates, though they do not all have the same number of links and turning-joints. As side-lines, two new inversion linkages and two other straight-line mechanisms are found. All of them are each other’s cognate and produce the same straight line. The presented cognates cover a wide range in sizes.

TECHNICAL BRIEFS

In tension-tension, as well as in compression-tension, fatigue testing it has been observed that there exist combinations of loading and specimen characteristics for which the specimen becomes unstable. The theory explaining this behavior is well known and stability curves which predict the stability or instability of the specimen have been tabulated. Unfortunately these tabulations do not extend into the regions of interest for actual fatigue tests. The present work outlines the theory and offers a simple criterion for the design of stable fatigue tests which results from the extension of these stability curves.

Axisymmetric and asymmetric vibrations of a membrane disk having a central hole and disk-central clamp friction are studied. It is shown that the vibration frequencies can be obtained by considering an equivalent disk having full central clamping.

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